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Derive the formula for the kinetic energy of an object of mass m, moving with velocity v.

Let us say an object of mass m is moving with speed v, so energy which the object has because of its motion or speed is its kinetic energy.

Now, let us assume object was initially at rest, some work was done on it to increase its speed to v and it gained kinetic energy.

Now from law of conservation of energy, we know kinetic energy will be equal to work done on the object.

Suppose F force was applied on an body of mass m , initially at rest to increase its speed to v in time t, in this time object also covered distance S.

We know work done W on a body of mass m, which moved a distance S due a force F applied on it is given as

W = F × s

Also, Force applied on the object of mass m moving with acceleration a is given as

F = ma

So, work done will be

W = ma× s

We have Newton’s second equation of motion which related displacement S covered by object in time t, having initial velocity u, with its acceleration a, equation is

Initially object is at rest so, initial velocity is

u = 0 m/s

so, equation becomes

So, putting value of displacement s, we get Work done as

This can be further simplified as

Using newton first equation of motion which relates final velocity v gained by object in time t, moving initially with initial velocity u, with acceleration of the object as

v = u + at

Initially object is at rest so, initial velocity is

u = 0 m/s

So, equation becomes

v = 0 + at = at

now, replacing the value of v in above equation to find the work done we get

Now this work done is equal to kinetic energy of the object of mass m moving with velocity v, i.e. Kinetic energy of the object is